308 



OF THE POSITIONS OF EQUILIBRIUM. 



which the conditions of the question will be truly satisfied, for only 

 one angle of the figure will fall below the plane of floatation. 



In order therefore to exhibit those positions, we shall suppose the 

 specific gravity of the floating prism to be expressed by 0.753, which 

 is the arithmetical mean between the limits above assigned ; then, by 

 operating according to the rules under equations (233) and (234), we 

 shall obtain _ 



tf 0.46703 ^4x28' 20 2 rz 24.429 as for- 



merly computed ; 



and after a similar manner, we have 



= 



a 9 ) = 



0.46703 S (4X28 9 20*) 



- 



1UUO 



=z2.528; 



consequently, by addition and subtraction, we shall get 

 x = m -f n 24.429 -f 2.528 = 26.957 inches, and x = m n = 



24.429 2.528 21.901 inches; 

 and the corresponding values oft/, are 

 21.901 and 26.957 inches respectively. 



390. The positions of equilibrium corresponding to the above values 

 of a: and y, are as represented 

 in the annexed diagrams, where 

 it may be shown that the 

 straight lines F E, F H and /D, 

 fi are equal to one another, 

 and also that the areas of the 

 immersed spaces E c H and D ci 

 are respectively to the whole 

 areas ABC and abc, as the 

 specific gravity of the solid, is to that of the fluid on which it floats, 

 or as 0.753 to unity in the case of water. 



These conditions being satisfied, the body will float in equilibrio in 

 the positions here exhibited; and it from hence appears, that the 

 problem admits of a complete solution, by retaining the specific gra- 

 vity of the solid within determinate limits. 



391 . When the transverse section of the floating prism, is in the 

 form of an equilateral triangle ; then a and b are equal to one another, 

 and equation (230) becomes 



_ b s cos id) \/ 3 b^ s* 



r $ ' 5 / 9 > 



